Primal and dual-primal iterative substructuring methods of stochastic PDEs

Abstract

A novel non-overlapping domain decomposition method is proposed to solve the large-scale linear system arising from the finite element discretization of stochastic partial differential equations (SPDEs). The methodology is based on a Schur complement based geometric decomposition and an orthogonal decomposition and projection of the stochastic processes using Polynomial Chaos expansion. The algorithm offers a direct approach to formulate a two-level scalable preconditioner. The proposed preconditioner strictly enforces the continuity condition on the corner nodes of the interface boundary, while weakly satisfying the continuity condition over the remaining interface nodes. This approach relates to a primal version of an iterative substructuring method. Next, a Lagrange multiplier based dual-primal domain decomposition method is introduced in the context of SPDEs. In the dual-primal method the continuity condition on the corner nodes is strictly satisfied while Lagrange multipliers are used to enforce continuity on the remaining part of the interface boundary. For numerical illustrations, a two dimensional elliptic SPDE with non-Gaussian random coefficients is considered. The numerical results demonstrate the scalability of these algorithms with respect to the mesh size, subdomain size, fixed problem size per subdomain, order of Polynomial Chaos expansion and level of uncertainty in the input parameters. The numerical experiments are performed on a Linux cluster using MPI and PETSc libraries.